Delay-differential equations have been studied extensively (see, e.g., [1]}), but little is known about their integrability properties or behavior in the complex domain. In [2]}, delay-differential equations are studied from these perspectives. Elliptic function solutions play an important role in this analysis: requiring the admittance of multi-parameter families of elliptic solutions was shown to be an effective tool to isolate equations of interest. Equations with such solutions are analogues of the Quispel–Roberts–Thompson (QRT) map [3]}, which underlies the integrability of discrete Painlevé equations. In the spirit of [3]}, we establish relationships between known semi-discrete integrable models and particular delay-differential equations with multi-parameter elliptic solutions via simple reductions.
